The authors analyze why flow matching models (which generate data by integrating a learned velocity field) need many integration steps. They split the velocity field's derivative into two parts: strain (how the flow stretches) and vorticity (how it rotates). Strain causes errors to grow exponentially during integration, while vorticity only adds linear error. They show optimal transport flows have zero vorticity and propose regularizing strain more heavily than vorticity, cutting integration error by 2.7× on synthetic data.
Main takeaways:
- Flow matching integration error comes mainly from strain (stretching), not vorticity (rotation)
- Optimal transport velocity fields are rotation-free and can be integrated exactly with simple Euler steps
- Weighted regularization that penalizes strain more than vorticity reduces the number of steps needed
- On CIFAR-10, lightweight fine-tuning with this regularization improves image quality by 14% at 10 integration steps
- The math is grounded in the Jacobian decomposition of the velocity field